[1003.5680]
A dynamical classification of the range of pair interactions

Authors:

Andrea Gabrielli, Michael Joyce, Bruno Marcos, Francois Sicard

Abstract:

We formalize and discuss the relevance of a classification of pair
interactions based on the convergence properties of the {\it forces} acting on
particles as a function of system size. We do so by considering the behavior of
the probability distribution function (PDF) P(F) of the force field F in a
particle distribution in the limit that the size of the system is taken to
infinity at constant particle density, i.e., in the "usual" thermodynamic
limit. For a pair interaction potential V(r) with V(r \to \infty) ~ 1/r^\gamma
defining a {\it bounded} pair force, we show that P(F) converges continuously
to a well-defined and rapidly decreasing PDF if and only if the {\it pair
force} is absolutely integrable, i.e., for \gamma > d-1, where d is the spatial
dimension. We refer to this case as {\it dynamically short-range}, because the
dominant contribution to the force on a typical particle in this limit arises
from particles in a finite neighborhood around it. For the {\it dynamically
long-range} case, i.e., \gamma </- d-1, on the other hand, the dominant
contribution to the force comes from the mean field due to the bulk, which
becomes undefined in this limit. We discuss also how, for \gamma </- d-1, P(F)
may be defined in a weaker sense, using a regularization of the force summation
which is a generalization of the so-called "Jeans swindle" employed to define
Newtonian gravitational forces in an infinite static universe. We explain that
the distinction of primary relevance in this context is, however, between pair
forces with \gamma > d-2 (or \gamma < d-2), for which the PDF of the {\it
difference in forces} is defined (or not defined) in the infinite system limit,
without any regularization.

This nice paper (unfortunately not crossposted to astro-ph) studies the well-definedness of classical interactions which go like a power-law in the long-range limit, in d dimensions. Newtonian gravity in three dimensions is a particular marginally pathological case. The fact that Newtonian gravity is not well-defined for an infinite system is of course well known in cosmology, but this puts the problem nicely in context. (It would be interesting to know what are the statistical properties of similar systems in general relativity, where there is no problem with infinite systems and the physics is quite different.)

As a minor issue, the authors also argue that systems where the absolute force diverges can be considered well-defined is the relative forces between particles remain finite, which seems to me odd, since absolute acceleration is measurable.